Algebra Prelim, August 1997

Answer all questions

1. Let R be a commutative ring with unity and let I, J be ideals of R.
   1. Prove that the product IJ = {x $\in$ R | x = S_{i = 1}ⁿa_ib_i with a_i $\in$ I and b_i $\in$ J} is an ideal of R.

   2. Prove that IJ $\subseteq$ I $\cap$ J.

   3. If I + J = R, prove that IJ = I $\cap$ J.

   4. If IJ = I $\cap$ J for all ideals of R and R is an integral domain, prove that R is a field. (Hint: let I = Ra where a$\ne$ 0 be a principal ideal or R.)

2. Let F be a finite Galois extension of the field K with Gal(F/K) $\cong$ S₅.
   1. Show that there are more than 40 fields strictly between F and K.

   2. Show that there is a unique proper subfield E of F with E$\ne$K such that E/K is a Galois extension. Determine [E : K] and describe Gal(E/K) up to isomorphism.

3. Let G be a group of order 455.
   1. Prove that G is not simple.

   2. Prove that G is cyclic.

4. Let R be a PID, let n be a positive integer, and let A and B be finitely generated R-modules. If Aⁿ $\cong$ Bⁿ, prove that A $\cong$ B. (Aⁿ denotes the direct sum of n copies of A.)

5. Let P be a finitely generated projective $\mathbb {Z}$-module. If P is also injective, prove that P = 0.

6. Let A, B be abelian groups, and let m be a positive integer. Prove that A $\otimes$ (B/mB) $\cong$ (A $\otimes$ B)/m(A $\otimes$ B).

7. Prove that a group of order 588 is solvable.

8. Let K = $\mathbb {Q}$($\sqrt{2}$ + $\sqrt{3}$,$\sqrt{5}$).
   1. Determine [K : $\mathbb {Q}$].

   2. Compute Gal(K/$\mathbb {Q}$).